With his ripped black jeggings, DMs, floppy Mohican, oversized T-shirt and underfed frame, the 1975's singer, Matt Healy, looks like a Spanish crustie all set to defend the last squat in Hackney. He jumps into the photographers' pit before the first song, The City, is over, slapping the hands of the front row; later, he'll turn up on the balcony for a run around. Most of the rest of his scrawny associates – all black vests, tattoos and sinew – look like fellow guttersnipe desperadoes.

The 1975 are, though, one of the biggest and shiniest new pop bands in the UK, the breakout act of 2013, who peddle an unlikely variant of 80s funk-pop cut with 21st-century R&B; a Mancunian all-boy version of Haim, perhaps. Pretty much everything they do is met with a boy band-calibre screech: the first clattery, processed bars of The City; the backwards-sounding intro into M.O.N.E.Y. – one of their finer R&B cuts – and its new, effects-laden live rock outro. When the band take more than a few seconds to reset between songs, they are urged on with yet more screams.

Technically, the 1975 should probably be called the 1985, or the 2005. But the band name actually comes from a date Healy found scrawled in a book, rather than any notional affiliation with the year of Patti Smith's Horses. They are a thoroughly "now" sort of band, putting musical eras through the internet mincer and skimming the fat off the top. Processed guitars? Nagging choruses? Don't mind if they do. Yearning, multi-tracked vocals? Likewise. (An actual overripe sax solo teleported from the early years of MTV polishes off Heart Out.) Throughout the gig, everyone – drummer George Daniel, bassist Ross MacDonald and guitarist Adam Hann – swaps comfortably between their instrument and its digital version: keyboards, drum pads. They're impressively slick – too slick, perhaps – but still coltish enough to be endearing. Healy's response to all this adulation is just to dance harder. Disappointingly, he doesn't say much beyond the usual script, thanking fans for their support; in interviews, Healy is full of the sort of unguarded chat that would have shortened the distance from the audience to the stage yet further.

That distance has only just sprung up, though. This bunch of Cheshire "stoner muppets" (Healy's words) have gone in just a year from playing the upstairs room of a pub called the Barfly in north London (capacity: 200-odd) to selling out three nights at Brixton Academy (total ticketing: nearly 15,000), giving the impression of a rapid rise. Their debut album went straight in at No 1 last September; they supported the Rolling Stones in Hyde Park. They've had screaming teenagers, Lindsey Lohan backstage at their New York gigs, the works. As ever, though, the 1975's overnight success took years. They started off as a more run-of-the-mill emo act, changed their name a few times, and were rejected by every major label, who found the 1975's refusal to settle on one genre fickle, rather than marketable. The scattershot nature of their slew of EPs across 2012 and 2013 only made sense when their hyper-produced album finally came out. This wasn't a guitar band, bucking the trend for electronic pop. The 1975 were – are – a guitar band whose surprisingly gritty vignettes of small-town twenty-nothing life come out in succinct, rhythmic and toothsome doses, like Bruno Mars playing the yacht rock of Thomas Mars's Phoenix. Chocolate, one of their best-known hits - irrepressibly bouncy tonight – is, in all likelihood, not about actual chocolate, but something more potent.

If the 1975 embody a bunch of contradictions, they are not exactly wrapped in an enigma. Healy's parents are household-name actors (Tim Healy, Denise Welch) whose divorce was played out in the tabloids; his mother went on to win Celebrity Big Brother in 2012. Healy's singing can be overly mannered, and his band over-thought, but he's got a real throat-tightener in Is There Somebody That Can Watch You, a song he plays alone at the keyboard, having tried to get the crowd to simmer down. The song is directed at his much younger brother, caught up in the tumult of the divorce and the sale of the house, his big brother forever on tour.

Sex, by contrast, is about sex – the contractual give-and-take of fooling around that resonates with anyone past puberty. It starts exactly the same way as LCD Soundsystem's All My Friends, which is amusing, given the cribbing LCD Soundsystem's James Murphy has done (Bowie, just for starters). If they are calculated, the 1975 are just the latest in a long line of tuneful magpies with no active sense of shame who have cunningly titled their singles after appealing things: Girls, Chocolate, The City, Sex. It really is no mystery why they are successful; but it's heartening to find that there is something more to the 1975 than funky surfaces.